‘Angry alien’ in packing puzzle shocks mathematicians

Easier packing

(Image: Sipa Press/Rex)

When it comes to packing, you can’t beat a mathematician. Centuries of work has gone into finding the most efficient ways to pack identical objects into the densest possible arrangements. But the latest experiment shows that a surprising amount of chaos lurks within our attempts to create order.

Sharon Glotzer at the University of Michigan in Ann Arbor and her colleagues started with three distinct shapes&colon; a cube, a 12-sided dodecahedron and a 4-sided pyramid, or tetrahedron. First they created variations of each by slicing off corners, edges or both, until they had more than 55,000 shapes. “We wanted to look at a large enough set of shapes that we could start to see some trends that you can’t get from looking at a couple of shapes,” says Glotzer.

The team then used computer simulations to create identical copies of each new shape and pack them in the most efficient way possible. Finally, they displayed the results for the three shape “families” as three-dimensional landscapes, with changes in height corresponding to packing efficiency.

The team thought small changes to a shape would only gradually affect how it packs, meaning each landscape should be made up of gentle hills. That is the case for the dodecahedron family, but the landscape is bumpy and chaotic for variants of cubes and tetrahedrons. The 3D tetrahedron landscape, shown below from multiple angles, is so weird that the researchers nicknamed it the angry alien, seen best at the bottom left.

(Image&colon; Elizabeth R. Chen, Daphne Klotsa et al)

Real-world objects are likely to have minor defects that change their shapes in similar ways. That means a better understanding of packing variations could be important for nanotech materials built from small particles, or for pharmaceuticals in which the density of a drug matters.

“I believe this research has tremendous value for designing new materials based on nanoparticles,” says Oleg Gang at Brookhaven National Laboratory in Upton, New York. “It gives us a complete understanding of how particles of different shapes pack in larger-scale organisations. Many practical applications, from catalysis to batteries, depend on particle packing.”

Tomaso Aste at University College London also thinks the results could prove useful, but he adds that they need to be double-checked. Packing is a difficult problem, he says, and different computer simulations might find better or worse packing densities for particular shapes, which could alter the landscape.